Читать книгу Multi-parametric Optimization and Control - Efstratios N. Pistikopoulos - Страница 22

1 Let be convex functions defined on a convex subset . Their summation(1.5) is convex, and if at least of one is a strictly convex function, then their summation is strictly convex.

2 Let a be a positive number and be a (strictly) convex function defined in a convex subset . Then the product is (strictly) convex.

3 Let be a (strictly) convex function defined in , and be an increasing convex function defined on the range of in . Then, the composite function defined in is a (strictly) convex function.

4 Let be convex functions defined on a convex subset . If these functions are bounded from above, their pointwise supremum(1.6) is a convex function on .

5 Let be concave functions defined on a convex subset . If these functions are bounded from below, their pointwise infimum(1.7) is a concave function on .